Hath smaller fleas that on him prey,
And these have smaller fleas that bite 'em,
And so proceed ad infinitum." Jonathan Swift
"Great fleas have little fleas upon their backs to bite 'em,
And little fleas have lesser fleas, and so ad infinitum.
And the great fleas themselves, in turn, have greater fleas to go on,
While these again have greater still, and greater still, and so on." Augustus De Morgan
An infinite dissection of the plane by logarithmic spirals.
There is a nice bistable perception in the animation: when I first see it, I percieve a wholistic pattern rotating clockwise and shrinking. But after a short time, I see half the contours as fixed, and the other half rotating (with no shrinking). Try fixating on a single edge point to experience the second percept.
The pattern is the basis of Orthologia trispiralis, and the pattern and transform is the basis for the animation Orthologia twist.
Here's the geeky description. It's a checkerboard division (8x2) of the plane by logarithmic spirals with a pitch of +/- pi/4 radians. The checkerboard is colored (orthogonal to the contours) by the first 2^3 = 8 elements of the Thue-Morse sequence, [0 1 1 0 1 0 0 1]. The relative phase between the two sets of opposite chirality spirals rotates through pi/2 radians; in this example the phase of one set is fixed.
[Broken link: A volume rendering derived from the animation, with the time axis transposed to a third spatial dimension.]
There are no restrictions on use of this image. Claiming to be the originator or owner, explicitly or implicitly, is bad karma. A link (if appropriate), a note to email@example.com, and credit are appreciated but not required.
Oh, now thanks so much! See, I happen to belong to the Morse Clan, as it goes, and it's always fascinating to see others of the name floating around. We aren't a well-connected family and seem to find ourselves where we least expect it. Thanks again, at any rate! All the best to you and your endeavours!
"On a surface of negative curvature having at least two different normal segments there exists a set of geodesics that are recurrent without being periodic and this set has the power of the continuum."
M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 22 (1921), 84-100.
Enjoyed a lot of your algorithmically generated stuff.
Yeah, its sounds like their functionality largely overlaps. So the images are rendered pointwize, wheras in mathematica they are generated based on primatives. btw, your avy is awesome! Ive been just staring at it for a while now....
Thanks. It takes a fair amount of thought to generate each image, which of course is true of any medium.
I've stared at it far too long also, but have found a trick that allows me to reliably switch percept: flicking my eyes between the center (rotating) and a fixed contour near the edge (fixed) induced the switch. Still I can't linger on one percept for more than several seconds before the other invades.